Mathematicians Are Making Major Breakthroughs In The Understanding Of Prime Numbers

A breakthrough has been made in the quest to prove something
called the twin primes conjecture.

The twin primes conjecture asserts that there are infinitely many
pairs of twin primes — prime numbers that have only one number
between them, like 11 and 13, or 17 and 19.

While the twin primes conjecture has not been proven,
mathematicians have, over the last several months, made
significant progress in showing that there are infinitely many
pairs of primes that are, at most, some fixed distance from each
other.

The prime numbers — numbers that are divisible only by themselves
and one — have always been a major subject of study in
mathematics.

One particular aspect of the primes that has fascinated
mathematicians throughout the centuries is their distribution —
where primes fall on the number line. We know that the prime numbers become
over all rarer as numbers get larger, but do primes sometimes
cluster together, or do the gaps between consecutive prime
numbers get larger and larger?

The extreme case of primes clustering together is exemplified in
the twin primes conjecture.

Most mathematicians have a sense that the twin primes conjecture
should be true — the positioning of the prime numbers appear to
be more or less random, even though on average the gaps between
primes get larger, and if one has an infinitely long list of
random odd numbers, we should have an infinite collection of
pairs in our list. If at some point, prime numbers are always
more than two numbers away from each other, we have a non-random
aspect to their distribution that goes against this intuition.

2013 has been an exciting year for this line of research. In May,
University of New Hampshire mathematician Yitang Zhang showed
that there are infinitely many pairs of primes that are no more
than 70,000,000 numbers away from each other. This was the first
time such an upper bound was established.

Now, James Maynard, a researcher at the University of Montreal,
has submitted a paper with a new approach. He has shown that
there are infinitely many pairs of primes no further than 600
numbers away from each other.

The speed with which this type of research has developed this
year is extremely impressive, and we are getting closer to a
better understanding of the building blocks of number theory.